偏微分方程IV

前言

　　要使我國的數(shù)學事業(yè)更好地發(fā)展起來，需要數(shù)學家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學家創(chuàng)造更有利的發(fā)展數(shù)學事業(yè)的外部環(huán)境，這主要是加強對數(shù)學事業(yè)的支持與投資力度，使數(shù)學家有較好的工作與生活條件，其中也包括改善與加強數(shù)學的出版工作?！　某霭娣矫鎭碇v，除了較好較快地出版我們自己的成果外，引進國外的先進出版物無疑也是十分重要與必不可少的。從數(shù)學來說，施普林格（springer）出版社至今仍然是世界上最具權威的出版社?？茖W出版社影印一批他們出版的好的新書，使我國廣大數(shù)學家能以較低的價格購買，特別是在邊遠地區(qū)工作的數(shù)學家能普遍見到這些書，無疑是對推動我國數(shù)學的科研與教學十分有益的事?！　∵@次科學出版社購買了版權，一次影印了23本施普林格出版社出版的數(shù)學書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎數(shù)學書5本，應用數(shù)學書6本與計算數(shù)學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學某方面的前沿，例如基礎數(shù)學中的數(shù)論、代數(shù)與拓撲三本，都是由該領域大數(shù)學家編著的“數(shù)學百科全書”的分冊。對從事這方面研究的數(shù)學家了解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數(shù)學類的書以“經(jīng)典”為主，應用和計算數(shù)學類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學家，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學獎”。這些大數(shù)學家的著作無疑將會對我國的科研人員起到非常好的指導作用?！　‘斎唬?3本書只能涵蓋數(shù)學的一部分，所以，這項工作還應該繼續(xù)做下去。更進一步，有些讀者面較廣的好書還應該翻譯成中文出版，使之有更大的讀者群。　　總之，我對科學出版社影印施普林格出版社的部分數(shù)學著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內容概要

This volume of the Encyclopaedia contains two contributions．In the first Yu.V.Egorov gives an accomnt of microlocal analysis as a tool for investigating partial differemial equations．This 113ethod has become increasingly important in the theory．of Hamiltonian systems in recent years．        The second survey written by V.Ya．1vrii treats linear hyperbolic equations and systems．The author states necessary and sufficiient conditions for C∞-and L2-well-posedness and he studies the analogous pmhlem in the comext ofGevrey classes．He also describes,the latest results in，the theory of mixed problems for hyperbolic operators and concludes with a list of unsolved problems．    Both parts coyer recent research in two important fields,which before was scattered in numerous joumals．The book will hence be of immense value to graduate students and researchers in partial differential equationS and theoretical physics。

書籍目錄

Chapter 1.Microlocal Properties of Distributions　2.Wave Front of Distribution.Its Functorial Properties　  2.1.Definition ofthe Wave Front　  2.2.Localization ofWave Front　  2.3.Wave Front and Singularities of One—Dimensional　  2.4.Wave Fronts of Pushforwards and Pullbacks of a　3.Wave Front and Operations on Distributions　　3.1 The Trace of a Distribution.Product of Distnritbiaul Eiuation    3.2.The Wave Front of the Solution of a Differential Eqution  3.3.Wave Fronts and Integral OperatorsChapter 2.Pseudodifferential Operators　1.Algebra ofPseudodifferential Operators　　1.1.Singular Integral Operators    1.2.The Symbol    1.3.Boundedness of Pseudodifferential Operators    1.4.Composition of Pseudodifferential Operators    1.5.The Formally Adjoint Operator    1.6.Pseudolocality.Microlocality    1.7.Elliptic Operators    1.8.Garding’S Inequality    1.9.Extension 0f the Class of Pseudodifferential Operators 　2.Invariance of the Principal SymboJ Under Canonical Transformations 　 2.1.Invariance Under the Change ofVariables.  　2.2 The Subprincipal Symbol  　2.3.Canonical Transformations　  2.4.An Inverse Theorem　3.Canonical Forms ofthe Symbol　  3.1.Simple Characteristic Points　  3.2.Double Characteristics　  3.3.The Complex-alued Symbol　  3.4.The Canonical Form of the Symbol in a Neighbourhood of the Boundary.　4.Various Classes of Pseudodifferential Operators    4.1.The Lm/pδClasses    4.2.The Lm/φ,φ Classes    4.3。The Weyl Operators　5.Complex Powers ofElliptic Operators    5.1.The Definition ofComplex Powers.    5.2.Thc Construction of the Symbol for the Operator Az    5.3.The Construction of the Kernel of the Operator Az    5.4.The ξ-Function ofan Elliptic Operator    5.5.The Asymptotics of the Spectral Function and Eigenvalues    5.6.Complex Powers of an Elliptic Operator with Boundary Conditions　6.Pseudodifferential Operators in IRn and Quantization  　6.1.The Analogy Between the Microlocal Analysis and the Quantization  　6.2.Pseudodifierential 0perators in RnChapter 3.Fourier Integral Operators　1.The Parametrix of the Cauchy Problem for Hyperbolic Equations 　 1.1.The Cauchy Problem for the Wave Equation　  1.2.The Cauchy Problem for the Hyperbolic Equation of an Arbitrary 0rder.    1.3.The Method of Stationary Phase　2.The Maslov Canonical Operator　  2.1.The MaslOV Index　  2.2.Pre.canonieal Operator　  2.3.The Canonical Operator　  2.4.Some Applications.　3.Fourier Integral Operators    3.1.The Oscillatory Integrals    3.2.The Local Definition of the Fourier Integral Operator  ……Chapter 4 The Propagation of SingularitiesChapter 5 Solvbility of (Pseudo)Differential EquationsChapter 6 Smoothness of Solutions of Differential EquationsChapter 7 Transformation of Boundary-Value ProblemsChapter 8 HyperfuctionsReferences

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